Some housekeeping (again), installing necessary packages.
list.of.packages <- c("igraph", "tidygraph", "ggraph")
new.packages <- list.of.packages[!(list.of.packages %in% installed.packages()[,"Package"])]
if(length(new.packages)) install.packages(new.packages)
rm(list.of.packages, new.packages)Introduction
Welcome to Module 2. This script is available as R-notebook (pretty) as well as executable jupyter notebook on kaggle here. HAve fun!
Networks… So what?
So, before we talk about networks, one thing upfront… why should we? I mean, they undeniably look pretty, don’t they?
Somehow, the visualization of networks fascinates the human mind (find a short TED talk on networks and how they depict our world here), and has even inspired an own art movement, networkism (see some examples here).
Yet, besides that, is there an analytical value for a data scientist to bother about networks?
Networks in R
There are a number of applications designed for network analysis and the creation of network graphs such as gephi and cytoscape. Though not specifically designed for it, R has developed into a powerful tool for network analysis.
Significant network analysis packages for R include the network, sna, and igraph package. In addition, Thomas Lin Pedersen has recently released the tidygraph package that leverage the power of igraph in a manner consistent with the tidyverse workflow. Even better, he tops it up with ggraph, a consistent ´ggplot2´-look-and-feel network visualization package.
R can also be used to make interactive network graphs with the htmlwidgets framework that translates R code to JavaScript. Cool implementations thereof are the vizNetwork and networkD3 packages.
As analytical tool, I will in this lab mostly use igraph. In terms of functions, it is pretty much equivalent to network, yet slightly more powerful, better integrated, and maintained. Since both packages have many of the same functions, better don’t load them both at once.
The Basic Structure of Networks
The Basic Jargon
First of all, what is a network? Plainly speaking, a network is a system of elements which are connected by some relationship. The vocabulary can be a bit technical and even inconsistent between different disciplines, packages, and software. The whole system is (surprise, surprise) usually called a network or graph. The elements are commonly referred to as nodes (system theory jargon) or vertices (graph theory jargon) of a graph, while the connections are edges or links. I will mostly refer to the elements as nodes, and their connections as edges.
Generally, networks are a form of representing relational data. This is a very general tool that can be applied to many different types of relationships between all kind of elements. The content, meaning, and interpretation for sure depends on what elements we display, and which types of relationships. For example:
- In Social Network Analysis:
- Nodes represent actors (which can be persons, firms and other socially constructed entities)
- Edges represent relationships between this actors (friendship, interaction, co-affiliation, similarity ect.)
- Other types of network
- Chemistry: Interaction between molecules
- Computer Science: The wirld-wide-web, inter- and intranet topologies
- Biology: Food-web, ant-hives
The possibilities to depict relational data are manifold. For example:
- Relations among persons
- Kinship: mother of, wife of…
- Other role based: boss of, supervisor of…
- Cognitive/perceptual: knows, aware of what they know…
- Affective: likes, trusts…
- Interaction: give advice, talks to…
- Affiliation: belong to same clubs, shares same interests…
- Relations among organizations
- As corporate entities
- Buy from / sell to, leases to, outsources to
- Owns shares of, subsidiary of
- Joint ventures, strategic alliances
- Via their members
- Personnel flows
- Interlocking directorates
- Personal friendships
- Co-memberships
- Relations other (non-social) entities
- Patents
- Patents citing other patents
- Co-occurrence of technological classes *Research fields
- through citations
- through people co-affiliated with fields *Sectors
- input-output relations
- Labor mobility *Technologies
- Patent IPC classes
- Semantic co-occurrence
- Patents
Note: Content matters! Each relation yields a different structure & has different effects. Theories might make sense on inter-personal, but not inter-organizational or non-social context.
The Data-Structure of Relational Data
Edgelist
MOst real world relational data is to be found in what we call an edge list, a dataframe that contains a minimum of two columns, one column of nodes that are the source of a connection and another column of nodes that are the target of the connection. The nodes in the data are identified by unique IDs. If the distinction between source and target is meaningful, the network is directed. If the distinction is not meaningful, the network is undirected (more on that later). So, every row that contains the ID of one element in column 1, and the ID of another element in column 2 indicates that a connection between them exists. An edge list can also contain additional columns that describe attributes of the edges such as a magnitude aspect for an edge. If the edges have a magnitude attribute the graph is considered weighted (more on that later). Below an example ofa minimal edge list created with the tibble() function.
edge_list <- tibble(from = c(1, 2, 2, 3, 4), to = c(2, 3, 4, 2, 1))
edge_listSometimes it is preferable to also create a separate node list. At its simplest, a node list is a data frame with a single column - which I will label as “id” - that lists the node IDs found in the edge list. The advantage of creating a separate node list is the ability to add attribute columns to the data frame such as the names of the nodes or any kind of groupings.
node_list <- tibble(id = 1:4, group = sample(letters[1:2], 4, replace = TRUE))
node_listAdjacency Matrix
A second popular form of network representation is the adjacency-matrix (also called socio-matrix). It is represented as a \(n*n\) matrix, where \(n\) stands for the number of elements of which their relationships should be represented. The value in the cell that intercepts row \(n\) and column \(m\) indicates if an edge is present (=1) or absent (=0).
Tip: Given an edgelist, an adjacency matrix can easily be produced by crosstabulating:
adj_matrix <- table(edge_list) %>% as.matrix()
adj_matrix## to
## from 1 2 3 4
## 1 0 1 0 0
## 2 0 0 1 1
## 3 0 1 0 0
## 4 1 0 0 0Generating a Graph Object in igraph
To create an igraph object from an edge-list data frame we can use the graph_from_data_frame() function, which is a bit more straight forward than network(). There are three arguments in the graph_from_data_frame() function: d, vertices, and directed. Here, d refers to the edge list, vertices to the node list, and directed can be either TRUE or FALSE depending on whether the data is directed or undirected. By default, graph.data.frame() treats the first two columns of the edge list and any remaining columns as edge attributes.
library(igraph)
g <- graph_from_data_frame(d = edge_list, vertices = node_list, directed = FALSE)
g## IGRAPH fd9f11e UN-- 4 5 --
## + attr: name (v/c), group (v/c)
## + edges from fd9f11e (vertex names):
## [1] 1--2 2--3 2--4 2--3 1--4Lets inspect the resulting object. An igraph graph object summary reveals some interesting informations.
- First, it tells us the graph-type: undirected
UN, or directedDN - Afterwards, the number of nodes (4), and edges (6)
- Followed by the node attributes (node level variables), which in this case are only their name (
attr: name (v/c)) - Lastly, a list of all existing edges. Note:
n--mindicates an undirected,n->man directed edge.
Lets take a look at the structure of the object:
glimpse(g[[1]])## List of 1
## $ 1: 'igraph.vs' Named int [1:2] 2 4
## ..- attr(*, "names")= chr [1:2] "2" "4"
## ..- attr(*, "env")=<weakref>
## ..- attr(*, "graph")= chr "fd9f11e3-c69f-11e8-9859-73db0ee6b157"We see, the object has a list-format, consisting of sepperate lists for every node, containing some attributes which are irrelevant now, and an edgelist for every node, capturing its ego-network (eg., .. ..- attr(*, "names")= chr [1:2] "2" "4")
We can also plot it to take a look. igraph object can be directly used with the plot() function. The results can be adjusted with a set of parameters we will discover later. It’s not super pretty, therefore we will later also explore more powerfull plotting tools for rgaphs. However, its quick&dirty, so lets take it like that for now.
plot(g)
Yeah, that’s the graph. We We can also use the adjacency matrix to create the same graph.
g <- graph_from_adjacency_matrix(adj_matrix, mode = "undirected")
g## IGRAPH fdc43a6 UN-- 4 4 --
## + attr: name (v/c)
## + edges from fdc43a6 (vertex names):
## [1] 1--2 1--4 2--3 2--4We can inspect and manipulate the nodes via V(g) (V for vertices, its graph-theory slang), and edges with E(g)
V(g)## + 4/4 vertices, named, from fdc43a6:
## [1] 1 2 3 4E(g)## + 4/4 edges from fdc43a6 (vertex names):
## [1] 1--2 1--4 2--3 2--4We can also use most of the base-R slicing&dicing.
V(g)[1:3]## + 3/4 vertices, named, from fdc43a6:
## [1] 1 2 3E(g)[2:4]## + 3/4 edges from fdc43a6 (vertex names):
## [1] 1--4 2--3 2--4Remember, it’s a list-object. So, if we just want to have the values, we have to use the double bracket [[x]].
V(g)[[1:3]]## + 3/4 vertices, named, from fdc43a6:
## name
## 1 1
## 2 2
## 3 3We can also use the $ notation.
V(g)$name## [1] "1" "2" "3" "4"Networks are coming…
So, lets get serious. Appropriate for the weather these days in Denmark, the theme is “winter is comming…”. Therefore, we will have some fun analysing the Game of Thrones data provided by Andrew Beveridge. It is a Character Interaction Networks for George R. R. Martin’s “A Song of Ice and Fire” saga (yes, we are talking about the books…). These networks were created by connecting two characters whenever their names (or nicknames) appeared within 15 words of one another in one of the books in “A Song of Ice and Fire.” The edge weight corresponds to the number of interactions. THis is a nice skill you will have afeter the second part of M2 on your own.
Build the graph
edges.cooc.all <- fread("../input/GoT/asoiaf-all-edges.csv", data.table = FALSE)
nodes.cooc.all <- fread("../input/GoT/asoiaf-all-nodes.csv", data.table = FALSE)
head(edges.cooc.all)So, that’s what we have, a classical edgelist, with id1 in column 1 and id2 in column2. Note, the edges are in this case weighted.I don’t like the sepperating “-” between in the names, lets get rid of them.
colnames(edges.cooc.all) <- tolower(colnames(edges.cooc.all))
edges.cooc.all %<>%
mutate(source = gsub("-", " ", source),
target = gsub("-", " ", target)) Ok, lets see how many characters we have overal.
edges.cooc.all %>%
select(-type) %>%
gather(x, name, source:target) %>%
n_distinct(.$name)## [1] 5646That’s a lot. We might want to restrict ourselfes first to a small set of the 50 most connected nodes.
chars.main <- edges.cooc.all %>%
select(-type) %>%
gather(x, name, source:target) %>%
group_by(name) %>%
summarise(sum_weight = sum(weight)) %>%
ungroup() %>%
arrange(desc(sum_weight)) %>%
top_n(50)
head(chars.main)So far so good, if we only go by edge weights, Seems as if Tyrion might make it…. my favorite anyhow. Lets reduce our edgelist to this main characters, just to warm up and keep the overview.
edges.cooc <- edges.cooc.all %>%
filter(source %in% chars.main$name & target %in% chars.main$name) %>%
select(source, target, weight)g <- graph_from_data_frame(d = edges.cooc, directed = FALSE)
g## IGRAPH fe2183d UNW- 50 402 --
## + attr: name (v/c), weight (e/n)
## + edges from fe2183d (vertex names):
## [1] Aemon Targaryen (Maester Aemon)--Grenn
## [2] Aemon Targaryen (Maester Aemon)--Jeor Mormont
## [3] Aemon Targaryen (Maester Aemon)--Jon Snow
## [4] Aemon Targaryen (Maester Aemon)--Mance Rayder
## [5] Aemon Targaryen (Maester Aemon)--Robert Baratheon
## [6] Aemon Targaryen (Maester Aemon)--Samwell Tarly
## [7] Aemon Targaryen (Maester Aemon)--Stannis Baratheon
## [8] Arya Stark --Bran Stark
## + ... omitted several edgesNote that this co-occurence network is weighted (number of co-occurence), and undirected.
is_weighted(g)## [1] TRUEis_directed(g)## [1] FALSEInspect the graph
Overal graph attributes
We already know from the summary, but we can also count the number of nodes and edges as follows:
# Count number of edges
gsize(g)## [1] 402# Count number of vertices
gorder(g)## [1] 50We can give the graph a first plot to see what happens there. It’s not pretty, but we will fine-tune it later.
plot(g,
edge.width = (E(g)$weight / max(E(g)$weight)) * 5 # Only argument we use so far for the weighted edges
)
IS really not pretty. But we will work on that later. However, we already see that some nodes are not connected (isolated), so lets drop them for our network analysis. Lets also get rid of edges with super small weight.
g <- delete_edges(g, E(g)[weight < 20]) # How to delete edges in a graph pbject
g <- delete_vertices(g, degree(g) == 0) # How to delete nodes in a graph pbjectEdges
Lets see a bit which options we have to select edges. We could for example use inc() to select only edges that include a certain node.
Note: Almost all edge and many furtehr graph functions to follow, have an argument called mode. Here, it doesn’t matter, since we ahve an undirected network. In case the edges are directed, we want to specify which mode to consider (mode = c("all", "out", "in", "total")). More on that at a later point.
E(g)[[inc("Daenerys Targaryen")]] ## + 6/189 edges from fe64c76 (vertex names):
## tail head tid hid weight
## 13 Barristan Selmy Daenerys Targaryen 3 9 83
## 68 Daenerys Targaryen Drogo 9 11 123
## 69 Daenerys Targaryen Hizdahr zo Loraq 9 16 96
## 70 Daenerys Targaryen Jorah Mormont 9 24 161
## 71 Daenerys Targaryen Quentyn Martell 9 49 58
## 72 Daenerys Targaryen Robert Baratheon 9 39 26We could also select them due to their characteristics
E(g)[[weight >= 150]]## + 11/189 edges from fe64c76 (vertex names):
## tail head tid hid weight
## 12 Arya Stark Sansa Stark 2 43 155
## 20 Bran Stark Hodor 4 17 209
## 26 Bran Stark Robb Stark 4 38 169
## 55 Cersei Lannister Joffrey Baratheon 8 21 173
## 65 Cersei Lannister Tyrion Lannister 8 47 209
## 70 Daenerys Targaryen Jorah Mormont 9 24 161
## 83 Eddard Stark Robert Baratheon 12 39 334
## 112 Jeor Mormont Jon Snow 20 23 174
## 123 Joffrey Baratheon Sansa Stark 21 43 222
## 126 Joffrey Baratheon Tyrion Lannister 21 47 219
## 134 Jon Snow Samwell Tarly 23 41 228Or plot their weight distribution
hist(E(g)$weight)
There is muchg more, but lets leave it with that for now.
Nodes
The node level gives us the opportunity to select (in this case) certain characters, their connectivity characteristics, and their enviroment.
First we look at different measures of degree-centrality, which are a set of measures incicating a certain type of structural importance in the network.
Degree centrality
The first one is the degree centralities is the degree centrality, which plainly means how well connected the node is (= Number of Edges). The version for the weighted networks (= Sum of edgeweigths) can be accessed with strength(). We can also directly check which node has the maximum degree.
degree(g) ## Aemon Targaryen (Maester Aemon) Arya Stark
## 3 9
## Barristan Selmy Bran Stark
## 5 14
## Brienne of Tarth Bronn
## 4 2
## Catelyn Stark Cersei Lannister
## 16 20
## Daenerys Targaryen Davos Seaworth
## 6 2
## Drogo Eddard Stark
## 2 18
## Edmure Tully Gregor Clegane
## 3 4
## Grenn Hizdahr zo Loraq
## 2 2
## Hodor Ilyn Payne
## 4 4
## Jaime Lannister Jeor Mormont
## 16 4
## Joffrey Baratheon Jojen Reed
## 19 3
## Jon Snow Jorah Mormont
## 13 3
## Loras Tyrell Luwin
## 6 6
## Lysa Arryn Mance Rayder
## 4 1
## Margaery Tyrell Meera Reed
## 5 3
## Melisandre Meryn Trant
## 3 3
## Myrcella Baratheon Petyr Baelish
## 3 10
## Pycelle Renly Baratheon
## 6 10
## Rickon Stark Robb Stark
## 6 17
## Robert Baratheon Rodrik Cassel
## 16 5
## Samwell Tarly Sandor Clegane
## 5 6
## Sansa Stark Stannis Baratheon
## 17 13
## Theon Greyjoy Tommen Baratheon
## 6 6
## Tyrion Lannister Tywin Lannister
## 25 9
## Quentyn Martell Varys
## 2 7which.max(degree(g))## Tyrion Lannister
## 47strength(g)## Aemon Targaryen (Maester Aemon) Arya Stark
## 234 512
## Barristan Selmy Bran Stark
## 226 1206
## Brienne of Tarth Bronn
## 225 160
## Catelyn Stark Cersei Lannister
## 766 1438
## Daenerys Targaryen Davos Seaworth
## 547 195
## Drogo Eddard Stark
## 151 1175
## Edmure Tully Gregor Clegane
## 128 102
## Grenn Hizdahr zo Loraq
## 121 138
## Hodor Ilyn Payne
## 333 112
## Jaime Lannister Jeor Mormont
## 862 277
## Joffrey Baratheon Jojen Reed
## 1343 223
## Jon Snow Jorah Mormont
## 1238 216
## Loras Tyrell Luwin
## 167 238
## Lysa Arryn Mance Rayder
## 179 112
## Margaery Tyrell Meera Reed
## 258 255
## Melisandre Meryn Trant
## 208 102
## Myrcella Baratheon Petyr Baelish
## 87 477
## Pycelle Renly Baratheon
## 193 448
## Rickon Stark Robb Stark
## 263 966
## Robert Baratheon Rodrik Cassel
## 1091 137
## Samwell Tarly Sandor Clegane
## 472 259
## Sansa Stark Stannis Baratheon
## 1059 771
## Theon Greyjoy Tommen Baratheon
## 239 337
## Tyrion Lannister Tywin Lannister
## 1694 392
## Quentyn Martell Varys
## 80 400which.max(strength(g))## Tyrion Lannister
## 47Well, again Tyrion it is…
Eigenvector Centrality
Again, the eigenvector. Eigenvector centrality is a very interesting measure. The idea is that nodes are the more important when they are connected to more important nodes. So, if you have cool friends, you are also supposed to be somewhat cooler. You already see, the calculation of such values in a network becomes circular, therefore we need some mathematical magic, which the eigenvector does. We will dig into this concept deeper at another point. We scale it, since these values otherwise tend to become very large.
Note: Here, also a weighted version is available. Again, the ? is your friend.
eigen_centrality(g, scale = TRUE)$vector %>% round(3)## Aemon Targaryen (Maester Aemon) Arya Stark
## 0.064 0.332
## Barristan Selmy Bran Stark
## 0.073 0.297
## Brienne of Tarth Bronn
## 0.126 0.183
## Catelyn Stark Cersei Lannister
## 0.415 0.952
## Daenerys Targaryen Davos Seaworth
## 0.039 0.060
## Drogo Eddard Stark
## 0.007 0.754
## Edmure Tully Gregor Clegane
## 0.067 0.072
## Grenn Hizdahr zo Loraq
## 0.035 0.008
## Hodor Ilyn Payne
## 0.079 0.086
## Jaime Lannister Jeor Mormont
## 0.547 0.113
## Joffrey Baratheon Jojen Reed
## 0.922 0.046
## Jon Snow Jorah Mormont
## 0.364 0.038
## Loras Tyrell Luwin
## 0.112 0.061
## Lysa Arryn Mance Rayder
## 0.121 0.047
## Margaery Tyrell Meera Reed
## 0.200 0.051
## Melisandre Meryn Trant
## 0.065 0.089
## Myrcella Baratheon Petyr Baelish
## 0.067 0.353
## Pycelle Renly Baratheon
## 0.156 0.267
## Rickon Stark Robb Stark
## 0.098 0.428
## Robert Baratheon Rodrik Cassel
## 0.755 0.044
## Samwell Tarly Sandor Clegane
## 0.127 0.186
## Sansa Stark Stannis Baratheon
## 0.723 0.367
## Theon Greyjoy Tommen Baratheon
## 0.091 0.270
## Tyrion Lannister Tywin Lannister
## 1.000 0.333
## Quentyn Martell Varys
## 0.004 0.345Betweenness Centrality
Lastly, the betweenness centrality. That is also an very interesting concept. This one is a measure of how much a node connects otherwise unconnected nodes with each others. This is measured by the amount of shortest paths leading through the node.
betweenness(g)## Aemon Targaryen (Maester Aemon) Arya Stark
## 0.0 19.0
## Barristan Selmy Bran Stark
## 93.0 17.5
## Brienne of Tarth Bronn
## 0.0 0.0
## Catelyn Stark Cersei Lannister
## 94.0 96.5
## Daenerys Targaryen Davos Seaworth
## 0.0 0.0
## Drogo Eddard Stark
## 0.0 138.0
## Edmure Tully Gregor Clegane
## 0.0 15.0
## Grenn Hizdahr zo Loraq
## 0.0 0.0
## Hodor Ilyn Payne
## 85.0 8.5
## Jaime Lannister Jeor Mormont
## 67.5 57.0
## Joffrey Baratheon Jojen Reed
## 63.0 0.0
## Jon Snow Jorah Mormont
## 116.0 47.5
## Loras Tyrell Luwin
## 57.5 133.0
## Lysa Arryn Mance Rayder
## 0.0 0.0
## Margaery Tyrell Meera Reed
## 0.0 0.0
## Melisandre Meryn Trant
## 19.0 0.0
## Myrcella Baratheon Petyr Baelish
## 1.0 7.0
## Pycelle Renly Baratheon
## 25.0 31.0
## Rickon Stark Robb Stark
## 73.0 98.5
## Robert Baratheon Rodrik Cassel
## 155.5 25.0
## Samwell Tarly Sandor Clegane
## 20.0 13.0
## Sansa Stark Stannis Baratheon
## 137.0 103.0
## Theon Greyjoy Tommen Baratheon
## 30.0 0.0
## Tyrion Lannister Tywin Lannister
## 309.0 24.0
## Quentyn Martell Varys
## 0.0 0.0Neighborhood of a Node
Lastly, we can look at the surrounding of a node, meaning the ones it is connected to, its neighborhood.
neighbors(g, 'Robert Baratheon')## + 16/50 vertices, named, from fe64c76:
## [1] Barristan Selmy Catelyn Stark Cersei Lannister
## [4] Daenerys Targaryen Eddard Stark Jaime Lannister
## [7] Joffrey Baratheon Jon Snow Petyr Baelish
## [10] Pycelle Renly Baratheon Sansa Stark
## [13] Stannis Baratheon Tyrion Lannister Tywin Lannister
## [16] VarysLikewise, we can look at the ego-network of a node. That means how many nodes are in a certain geodesic distance. Plainly speaking, how many nodes are not more than x-steps away.
ego(g, 2, "Drogo")[[1]]## + 8/50 vertices, named, from fe64c76:
## [1] Drogo Daenerys Targaryen Jorah Mormont
## [4] Barristan Selmy Hizdahr zo Loraq Robert Baratheon
## [7] Quentyn Martell Tyrion LannisterWe can also not only look at it, but produce a new sub-graph only of this ego-network.
g.drogo <- make_ego_graph(g, 2, nodes = "Drogo")[[1]]
g.danny <- make_ego_graph(g, 2, nodes = "Daenerys Targaryen")[[1]]plot(g.drogo)
plot(g.danny)
Well, Danny seems to be way more connected than her husband, right?
Btw: To merge two graphs, just do:
g.merge <- g.drogo + g.danny
g.merge## IGRAPH fee7f58 UN-- 21 82 --
## + attr: name (v/c), weight_1 (e/n), weight_2 (e/n)
## + edges from fee7f58 (vertex names):
## [1] Stannis Baratheon--Tywin Lannister
## [2] Renly Baratheon --Stannis Baratheon
## [3] Pycelle --Varys
## [4] Petyr Baelish --Varys
## [5] Petyr Baelish --Sansa Stark
## [6] Petyr Baelish --Pycelle
## [7] Jon Snow --Stannis Baratheon
## [8] Joffrey Baratheon--Varys
## + ... omitted several edges(Global) Network structure
Finally, it is often also informative to look at the overal characteristics of the network. We will do this in more detail enxt time, but just so you know:
The density of a measure represents the share of all connected to all possible connections in the network
edge_density(g)## [1] 0.1542857Transistivity, also called the Clustering Cefficient indicates how much the network tends to be locally clustered. That is measured by the share of closed triplets. Again,w e will dig into that next time.
transitivity(g)## [1] 0.4552807The diameter is the longest of teh shortest paths between two nodes of the network.
diameter(g, directed = F, weights = NA)## [1] 4Finally, the mean distance, or average path lenght represents the mean of all shortest paths between all nodes. It is a measure of diffusion potential within a network.
mean_distance(g, directed = F)## [1] 2.337959Your turn
Ahh, you saw it comming, right? What about you explore the GoT network a bit on your own HERE. Lets see how that works out!
Directed Networks are comming…
So far so good, up to now we considered undirected networks, constructed by the amount characters co-occur. However, as you already might guess, that’s not where we stop. SOme general infos on directed networks:
- Relations do not necessarily have to be symmetric. Rather, often we find a logical or empirical direction in role based, affective and interactive relationships.
- Eg. is supervisor of…, gives advice to…, is supplier of…, invests in …
- This direction is often crucial for the correct interpretation
- Introduces new measurements, eg. the differentiation of degrees into out- (sending a tie) and in-degrees (receiving a tie)
An obvious example at the GoT case are family ties. Here, I will ose the nicely compiled dataset of the wonderful Shirin that can be found here. It contains a nodelist with house-affiliations and furtehr characteristics of main characters, and a edgelist of their family relationships.
Load the graph
rm(chars.main, g, g.danny, g.drogo, g.merge)
edges.fam <- readRDS("../input/GoT/union_edges.RDS")
nodes.fam <- readRDS("../input/GoT/union_characters.RDS")head(nodes.fam)head(edges.fam)Lets construct teh graph. We now also feed in a set of nodes with their characteristics.
g <- graph_from_data_frame(edges.fam,
vertices = nodes.fam,
directed = TRUE)
g## IGRAPH ff1ceaf DN-- 208 404 --
## + attr: name (v/c), male (v/n), culture (v/c), house (v/c),
## | popularity (v/n), house2 (v/c), color (v/c), shape (v/c), type
## | (e/c), color (e/c), lty (e/c)
## + edges from ff1ceaf (vertex names):
## [1] Lysa Arryn ->Robert Arryn
## [2] Jasper Arryn ->Alys Arryn
## [3] Jasper Arryn ->Jon Arryn
## [4] Jon Arryn ->Robert Arryn
## [5] Cersei Lannister ->Tommen Baratheon
## [6] Cersei Lannister ->Joffrey Baratheon
## + ... omitted several edgesVisualizing networks I
How does the plot look?
plot(g,
vertex.color = "blue",
vertex.shape = "circle")
Well, thats some uninformative hairball. Such things often happen when we want to plot large networks. This is a good point to start using the parameters in the igraph plotting function. Next time I will introduce you to my current favorite, ggraph. However, the igraph plotting function is relatively easy to sue. so its worth (if only for the sake of illustration) to tweak through all the parameters to get THE GoT family graph, as pretty as possible.
For plotting the legend, I am summarizing the edge and node colors upfront. You will later see why.
color_vertices <- nodes.fam %>%
group_by(house, color) %>%
summarise(n = n()) %>%
filter(!is.na(color))
colors_edges <- edges.fam %>%
group_by(type, color) %>%
summarise(n = n()) %>%
filter(!is.na(color))Ok, we are ready, lets do this plot!
plot(g,
layout = layout_with_fr(g), # graph layout, more on that later
vertex.label = gsub(" ", "\n", V(g)$name), # Just altered the label that it is in two lines
vertex.shape = V(g)$shape, # load the predefined shape of the nodes (male/female)
vertex.color = V(g)$color, # load the predefined color of the nodes (houses)
vertex.size = (V(g)$popularity + 0.5) * 5, # define node-size by popularity
vertex.frame.color = "gray", # color of the frame around the node
vertex.label.color = "black", # color of the label
vertex.label.cex = 0.8, # size of the label
edge.arrow.size = 0.5, # size of the arrow
edge.color = E(g)$color, # laod predefined edge/color (relationship)
edge.lty = E(g)$lty) # Edge style
legend("topleft", legend = c(NA, "Node color:", as.character(color_vertices$house), NA, "Edge color:", as.character(colors_edges$type)), pch = 10,
col = c(NA, NA, color_vertices$color, NA, NA, colors_edges$color), pt.cex = 3, cex = 2, bty = "n", ncol = 1,
title = "")
legend("topleft", legend = "", cex = 3, bty = "n", ncol = 1,
title = "Game of Thrones Family Ties")
What a beauty! Thanks to Shirin’s nice upfront work, it was rather easy to plot it that nicely. However, we could also define our colors as we wished. Here the R-Color Palettes might become handy.
Community Detection
You might have already guessed, we can very well also do a clustering exercise in networks. Here, we do not cluster nodes according to their similarity in attributes, but according to their connectivity. There are plenty of algorithms,, and we will explore further ones lateron. Most of them aim to find communities with maximum within/connectivity, and minimum between/connectivity.
However, most of them are not designed to work with directed networks. Therefore, we will convert our nice network for now to an undirected one.
g.ud <- as.undirected(g)First, we will give it a try with the edge-betweenness algorithm (Newman-Girvan). Here, high-betweenness edges are removed sequentially (recalculating at each step) and the best partitioning of the network is selected.
Lets take a look how it works.
?cluster_edge_betweennessAnd we now run it. As an hirarchical community detection technique. Since it is an hirarchical one, we can again plot a dendogram which we already know from the hirarchical clustering
ceb <- cluster_edge_betweenness(g.ud)
dendPlot(ceb, mode="hclust")
plot(ceb, g.ud,
vertex.frame.color = V(g)$color, # load the predefined color of the nodes (houses)
vertex.size = (V(g)$popularity + 0.5) * 5 # define node-size by popularity)
)
Lets only see how good it performs on the major houses, the rest is too small anyhow
bind_cols(com = ceb$membership, house = V(g.ud)$house) %>%
group_by(house) %>%
filter(n() >= 10) %>%
ungroup() %>%
table()## house
## com House Baratheon House Frey House Greyjoy House Lannister
## 1 0 0 0 0
## 2 9 0 0 4
## 3 1 0 0 0
## 4 0 13 0 0
## 5 0 7 0 9
## 6 0 0 14 0
## 7 0 0 0 13
## 8 0 0 0 0
## 9 0 0 0 0
## 10 0 0 0 0
## 11 0 0 0 0
## 12 0 0 0 0
## house
## com House Martell House Stark House Targaryen House Tyrell
## 1 0 7 0 0
## 2 0 0 0 1
## 3 2 0 13 0
## 4 0 0 0 0
## 5 0 0 0 0
## 6 0 0 0 0
## 7 0 0 0 0
## 8 11 0 0 0
## 9 0 6 0 0
## 10 0 13 0 0
## 11 0 0 0 9
## 12 0 0 0 5We see, indeed, that the communities for the most part capture the affiliation to the great houses.
Your turn
AGain, its time to have some fun on your own. HERE you will find another kaggle notebook where you can demonstrate your network analysis skills even more!